[Paper Review] An Economic Bubble Model and Its First Passage Time

Angelos Dassios, Luting Li|arXiv (Cornell University)|Mar 21, 2018

Complex Systems and Time Series Analysis20 references3 citations

TL;DR

This paper proposes a novel time-homogeneous diffusion process with exponential decay in the drift term to model economic bubbles, enabling explicit computation of the first passage time (FPT) density for crash prediction. The model's key contribution is a closed-form approximation for the downward FPT density via perturbation techniques, validated through calibration on the dot-com bubble, Chinese stock crash, and Bitcoin price dynamics, showing strong predictive accuracy for crash probabilities.

ABSTRACT

We introduce a new diffusion process Xt to describe asset prices within an economic bubble cycle. The main feature of the process, which differs from existing models, is the drift term where a mean-reversion is taken based on an exponential decay of the scaled price. Our study shows the scaling factor on Xt is crucial for modelling economic bubbles as it mitigates the dependence structure between the price and parameters in the model. We prove both the process and its first passage time are well-defined. An efficient calibration scheme, together with the probability density function for the process are given. Moreover, by employing the perturbation technique, we deduce the closed-form density for the downward first passage time, which therefore can be used in estimating the burst time of an economic bubble. The object of this study is to understand the asset price dynamics when a financial bubble is believed to form, and correspondingly provide estimates to the bubble crash time. Calibration examples on the US dot-com bubble and the 2007 Chinese stock market crash verify the effectiveness of the model itself. The example on BitCoin prediction confirms that we can provide meaningful estimate on the downward probability for asset prices.

Motivation & Objective

• To develop a tractable, time-homogeneous diffusion process that realistically models asset price dynamics during economic bubble cycles.
• To address the lack of explicit first passage time density estimation in existing bubble models, particularly for crash timing.
• To reduce parameter dependence and mitigate overfitting by introducing a scaling factor in the drift term based on exponential decay of the price.
• To provide a practical calibration scheme grounded in economic features for real-world application.
• To enable probabilistic forecasting of bubble burst times using perturbation-based analytical solutions.

Proposed method

• Proposes a new SDE with a drift term that incorporates mean reversion via exponential decay of the scaled price, reducing dependence on model parameters.
• Establishes the process as a strong, unique, and recurrent strong Markov process with a well-defined stationary distribution.
• Derives the Laplace transform of the first passage time (FPT) and applies perturbation techniques to obtain a closed-form approximation for the downward FPT density.
• Calibrates the model using historical price data by mapping observed price drops to hitting levels in log-price space and estimating parameters via economic features.
• Employs Monte Carlo simulation (10,000 paths) to estimate the perturbation error and validate the accuracy of the FPT density approximation.
• Transfers parameters from an extended SDE to a standard SDE form to enable analytical tractability and use of Corollary 4.9 for FPTD computation.

Experimental results

Research questions

• RQ1 Can a simple, time-homogeneous diffusion process with exponential decay in the drift term effectively model the dynamics of an economic bubble?
• RQ2 Does the inclusion of a scaling factor in the drift term reduce parameter dependence and improve model robustness?
• RQ3 Can the first passage time density for downward price movement be derived in closed form using perturbation techniques?
• RQ4 How accurately can the model predict the probability of price drops (e.g., 10%, 20%, 30%) in real-world bubble scenarios?
• RQ5 Can the model provide meaningful crash time estimates for speculative assets like Bitcoin, even before a crash occurs?

Key findings

• The proposed diffusion process is a well-defined semimartingale with a strong and unique solution, and its first passage time is almost surely finite.
• The stationary distribution of the process has a neat functional form, supporting long-term stability and calibration.
• The perturbation-based method yields a closed-form approximation for the downward first passage time density with relative errors below 2% for 10%–50% price drops.
• For Bitcoin (Dec 2017–Jan 2018), the model predicted a 69.38% probability of a 10% drop and a 17.87% probability of a 30% drop, both consistent with observed data.
• The model estimated a 40.19% probability of a 20% drop in Bitcoin, indicating a more than 50% chance the price would remain above 11,497.30, suggesting no imminent collapse.
• Validation on the US dot-com bubble and 2007 Chinese stock market crash confirmed the model’s effectiveness in capturing historical bubble dynamics and crash timing.